Wave propagation in the frequency regime in one-dimensional quasiperiodic media -Limiting absorption principle

TL;DR

This paper investigates the Helmholtz equation with quasiperiodic coefficients in one dimension, applying the limiting absorption principle with Robin boundary conditions to ensure well-posedness and develop a numerical solution method.

Contribution

It introduces a modified boundary condition approach for the limiting absorption principle in quasiperiodic media and proves its validity under certain frequency conditions.

Findings

The limiting absorption principle holds for quasiperiodic Helmholtz equations under specific assumptions.

Robin-to-Robin boundary conditions are necessary for the absorption limit.

A numerical method for computing the physical solution is proposed.

Abstract

We study the one-dimensional Helmholtz equation with (possibly perturbed) quasiperiodic coefficients. Quasiperiodic functions are the restriction of higher dimensional periodic functions along a certain (irrational) direction. In classical settings, for real-valued frequencies, this equation is generally not well-posed: existence of solutions in L 2 is not guaranteed and uniqueness in L $\infty$ may fail. This is a well-known difficulty of Helmholtz equations, but it has never been addressed in the quasiperiodic case. We tackle this issue by using the limiting absorption principle, which consists in adding some imaginary part (also called absorption) to the frequency in order to make the equation well-posed in L 2 , and then defining the physically relevant solution by making the absorption tend to zero. In previous work, we introduced a definition of the solution of the equation with absorption based on Dirichlet-to-Neumann (DtN) boundary conditions. This approach offers two key advantages: it facilitates the limiting process and has a direct numerical counterpart. In this work, we first explain why the DtN boundary conditions have to be replaced by Robin-to-Robin boundary conditions to make the absorption go to zero. We then prove, under technical assumptions on the frequency, that the limiting absorption principle holds and we propose a numerical method to compute the physical solution.